In this work, we utilize the Petrov-Galerkin conditions [1]} and relaxed greedy index selection technique [2]}, [3]}, and present several new relaxed greedy deterministic row and column methods. The paper is organized as follows. In the rest of this section, we give some notation. In Section  we briefly recap the classical relaxed greedy randomized row and column methods for solving the linear system. In Section  we present two relaxed greedy deterministic row and column methods together with their convergence theories. In Section  some numerical examples are provided to demonstrate the theoretical results.
